# Demand Systems

 First we derive an important formulation of the structural demand system mathematically, without bothering about economic interpretation. The next section will be more extensive, and includes the derivation of several important economic properties of demand systems. Therefore section IV.I.1 may be skipped by those who are only interested in the formulation of demand systems. Readers who are interested in the "whole story" may directly proceed to the next section. Theorem: If consumers are utility maximizers with utility function U(q) (which is twice differentiable) subject to a budget restriction p'q = m (q is a consumption vector, p is an exogenous price vector, and m is the exogenous budget) then an approximation to the consumers demand equation is given by (IV.I-1) Proof: is quite simple because it only involves optimizing the relevant Lagrangian function, yielding (IV.I-2) The second order conditions may be assumed to be satisfied (see section IV.I.1). On approximating (IV.I-2) we obtain (in differential form) (IV.I-3) or (IV.I-4) where H* is the Hessian matrix of the utility function U(q*). Now consider the following definitions (for the sake of convenience) (IV.I-5) and the following specification of consumer demand (as a function of m and p) (IV.I-6) or equivalently (IV.I-7) By combining (IV.I-4) and (IV.I-7) we find (IV.I-8) Since H is a symmetric matrix, it follows that (IV.I-9) is also symmetric. On substituting (IV.I-9) into (IV.I-8) we find (IV.I-10) From (IV.I-9), and (IV.I-10) it follows that (IV.I-11) (for details see next section). Now the structural differential demand system is obtained by partial substitution of (IV.I-10) into (IV.I-11) combined with (IV.I-7) (IV.I-12) Rewriting a little further (using (IV.I-10) and (IV.I-11)) yields the desired result (IV.I-1 repeated) (Q.E.D.).

© 2000-2022 All rights reserved. All Photographs (jpg files) are the property of Corel Corporation, Microsoft and their licensors. We acquired a non-transferable license to use these pictures in this website.
The free use of the scientific content in this website is granted for non commercial use only. In any case, the source (url) should always be clearly displayed. Under no circumstances are you allowed to reproduce, copy or redistribute the design, layout, or any content of this website (for commercial use) including any materials contained herein without the express written permission.

Information provided on this web site is provided "AS IS" without warranty of any kind, either express or implied, including, without limitation, warranties of merchantability, fitness for a particular purpose, and noninfringement. We use reasonable efforts to include accurate and timely information and periodically updates the information without notice. However, we make no warranties or representations as to the accuracy or completeness of such information, and it assumes no liability or responsibility for errors or omissions in the content of this web site. Your use of this web site is AT YOUR OWN RISK. Under no circumstances and under no legal theory shall we be liable to you or any other person for any direct, indirect, special, incidental, exemplary, or consequential damages arising from your access to, or use of, this web site.

Contributions and Scientific Research: Prof. Dr. E. Borghers, Prof. Dr. P. Wessa
Please, cite this website when used in publications: Xycoon (or Authors), Statistics - Econometrics - Forecasting (Title), Office for Research Development and Education (Publisher), http://www.xycoon.com/ (URL), (access or printout date).